Using renormalization theory, Zisook has shown that at small values of the dissipation (e.g., for a planar map of constant Jacobian B=0.999) the apparent rate of convergence, δ, of the parameter values for period doubling descends (‘‘crosses over’’) from its conservative value (δ=8.721 097 2. . .) to its dissipative value (δ=4.669 201 6. . .) as we consider higher and higher bifurcations. Moreover, this apparent δ at the nth bifurcation is a universal function, depending only on &. Employing a technique of Ghendrih and renormalization theory, I obtain an analytical expression for this function and show that it does not descend monotonically but has a minimum at Be,min≃3×10-6. I also analytically calculate the (orbit) scaling factor α at Be=0 and at Be=1 using some results of a second-order renormalization equation which I derive here. Finally, I obtain an analytical expression for the bifurcation values Cn(B) of a particular map (the Hénon map) as a function of n and B.
|Journal||Physical review A: Atomic, molecular, and optical physics|
|Publication status||Published - 1985|